Characterization of Total Graphs

Abstract

This thesis delves into the characterization and properties of regular total graphs and their chromatic numbers. A regular total graph is a unique graph structure where each vertex has the same degree, and the total graph, denoted as T(G), incorporates both the vertices and edges of a given graph G as vertices, with edges in T(G) representing adjacency or incidence relationships in G. The primary aim of this research is to elucidate the structural characteristics and chromatic properties of these regular total graphs. We consider “ordinary” graphs; that is, finite un-directed graphs with no loops or multiple edges. The total graph T(G) of a graph G is that graph whose vertex set is V(G)U E(G) and in which two vertices are adjacent if and only if they are adjacent or incident in G. A characterization of regular total graphs as well as some other properties of total graphs have been considered before. In this article we consider nonregular graphs and yield a method which enables us actually to determine whether or not they are total. Let G be an ordinary graph (finite, undirected, with no loops or multiple lines). Besides the chromatic number X(G) and line chromatic number X 0 (G), there is associated with G another positive integer X 00 (G), called the total chromatic number of G, which is the minimal number of colours required for colouring the elements (points and lines) of G such that no two elements which are either adjacent or incident have the same colour. The chromatic number, a fundamental parameter in graph theory, represents the minimum number of colors required to color the vertices of a graph such that no two adjacent vertices share the same color. This research aims to depend the understanding of how chromatic properties manifest in complete graphs and regular graphs. Regular graphs, the relationship between vertex degree and chromatic number is scrutinized, with new bounds and exact values established for specific classes of regular graphs. For complete graphs, the study reaffirms that the chromatic number is equal to the number of vertices, providing a clear and concise analysis of this well-known result. Closed loop cycle paths, also known as cycles, are fundamental structures in graphs where a sequence of vertices is connected by edges forming a loop with no repeats except for the starting and ending vertex. Understanding these cy- cles is crucial for various applications in computer science, network theory, and optimization problems. The thesis then explores different types of cycles, such as Hamiltonian and Eulerian cycles, and their significance in both directed and undirected graphs. Automorphisms and isomorphisms are fundamental concepts in graph theory, playing a crucial role in understanding the symmetry and equivalence of graph- s. Automorphisms, which are graph isomorphisms from a graph to itself, are explored in detail to understand the symmetries within a graph. Isomorphisms, which are bijections between vertex sets of two graphs that preserve adjacency, are analyzed to determine graph equivalence.